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Poisson laws and exterior stability for random alternating tensors

  • Pakin Methawisal
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Abstract

For fixed k≥3, we determine the critical law of totally isotropic r-spaces for a uniform random map ΘN:Λk𝔽qN→𝔽qm. At the exact balance mbinom rk=r(N−r), the entire null configuration is asymptotically an independent Bernoulli subset of Gr(r,𝔽qN) in total variation, uniformly in q and m. Consequently, the counting measure Ξr is asymptotically a Poisson point process, and its total mass XN,r is asymptotically Poisson. An exterior-rank stability theorem shows that near-extremal families decompose into Grassmann clusters with uniformly controlled span deficiency. We obtain quantitative rates and identify the first exterior-dependence scale. We also show that rare null (r+1)-spaces force high-order factorial-moment divergence, while the fixed-target bilinear cases m=1,2 exhibit non-Poisson critical behavior.

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Cite

BibTeXRIS
BibTeX
@misc{methawisal2026poisson,
  title = {{Poisson laws and exterior stability for random alternating tensors}},
  author = {Pakin Methawisal},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14340},
  eprint = {2609.14340},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14340},
}
RIS
TY  - UNPB
TI  - Poisson laws and exterior stability for random alternating tensors
AU  - Pakin Methawisal
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14340
UR  - https://arxiv.org/abs/2609.14340
AB  - For fixed k≥3, we determine the critical law of totally isotropic r-spaces for a uniform random map Θ_N:Λ^k𝔽_q^N→𝔽_q^m. At the exact balance mbinom rk=r(N−r), the entire null configuration is asymptotically an independent Bernoulli subset of Gr(r,𝔽_q^N) in total variation, uniformly in q and m. Consequently, the counting measure Ξ_r is asymptotically a Poisson point process, and its total mass X_N,r is asymptotically Poisson. An exterior-rank stability theorem shows that near-extremal families decompose into Grassmann clusters with uniformly controlled span deficiency. We obtain quantitative rates and identify the first exterior-dependence scale. We also show that rare null (r+1)-spaces force high-order factorial-moment divergence, while the fixed-target bilinear cases m=1,2 exhibit non-Poisson critical behavior.
ER  -